To understand how solar eclipses are predicted, we first must define some terminology relating to various types of "months". These months are not calendar months, but rather are time intervals related to the moon's orbit.
Synodic Month (29.53 days): This is the most common period referenced in the context of the moon. It measures the time it takes for the Moon to complete a revolution with respect to the line between the Earth and the Sun, or the time between two full moons or two new moons. However, this month is not equal to the period of the Moon's orbit, because the line between the Earth and the Sun shifts, as the Earth itself revolves.
The above image shows the position of the Earth and the Moon relative to the Sun at two successive new moons. Note the angle between the line connecting the Earth and Sun and the line marking the point at which the Moon has completed exactly one orbit around the Earth. Such an orbit takes 27.32 days.
Draconic Month (27.21 days): This "month" is the average time between successive crossings of the ascending node by the Moon on its orbit. The nodes of an orbit are the orbit's intersection with some plane of reference. In the context of eclipses, we are concerned with the intersection of the Moon's orbit around the Earth with the plane containing the Earth's orbit and the Sun. Since the Moon's orbit is inclined to this plane, there are only two points of intersection (see diagram below).
The two nodes of the Moon's orbit are the ascending node, at which point the Moon crosses from below the plane of the Earth's orbit to above (where the area "above" the plane is the space beyond the Earth's north pole), and the descending node, where the opposite occurs. Since, due to the effect of the Sun's gravity, the nodes of the Moon's orbit shift with time against the Moon's direction of orbit, the draconic month (time between crossing ascending node twice) is slightly shorter than the orbital period of the Moon (27.32 days). The draconic month is important for the prediction of eclipses because eclipses can only occur when the Moon is in the plane of the Earth's orbit and the Sun, i.e., when the Moon is at an ascending or descending node.
Anomalistic Month (27.55 days): The Moon's orbit, just like the Earth's and that of other celestial bodies, is not a perfect circle. Thus the Moon's distance from Earth varies. The farthest point of the Moon's orbit from the Earth is known as the apogee (deriving from Ancient Greek "from" and "earth", the second of which also gives us geography, geocentric, etc.), and the closest point the perigee (with "peri" meaning "around").
|Diagram of the Moon's orbit (elongation exaggerated)|
The anomalistic month measures the time it takes for the Moon to travel from one apogee (or perigee) to the next. Note that this length of time is slightly longer than a lunar orbital period, because the apogee and perigee move along the lunar orbit in the same direction as the Moon's motion, so it takes longer for the Moon to "catch up" to its apogee than for the Moon to simply complete an orbit. The anomalistic month is important for eclipse prediction, because although it does not affect the apparent position of the Moon in the sky, whether the Moon is closer to apogee or perigee does affect its apparent size, and thus can affect the type and duration of an eclipse.
|Apparent size of the Moon at apogee and perigee.|
Note that the apparent size of the Sun varies too, as the Earth's orbit is also elliptical, but the variation occurs at only a fraction of the magnitude (about 3% variation versus the Moon's 12%). Therefore, we focus on the Moon's apparent angular diameter as the determining factor.
In summary, progress through the synodic month indicates the Moon's position relative to the Earth-Sun line (in the Earth-Sun orbital plane), progress through the draconic month indicates how far the Moon is from the Earth-Sun orbital plane, and progress through the anomalistic month indicates the Moon's apparent size. It so happens that the length of time equal to 223 synodic months is about the same as that of 242 draconic months and 239 anomalistic months, all of which equal approximately 6585.32 days. The reason this is significant is that after this period of time passes, a whole number multiple of each of the three "months" will have passed, and the Moon will be in almost exactly the same position relative to the Sun, the same apparent size, and the same inclination from the Earth-Sun orbital plane. Thus if an eclipse occurs at a given date, a nearly identical eclipse will occur again 6585.32 days later! A period of 6585.32 days, about 18 years, is known as a saros cycle.
The sequence of eclipses produced beginning at a given date and including all the eclipses separated from the first by some number of saros cycles is known as a saros series. Eclipses, both lunar and solar, are classified by saros series. The next post explores the details and applications of saros cycles and series.
Sources: http://www.hermit.org/eclipse/Graphics/diagrams/Rotations3.png, http://en.wikipedia.org/wiki/Saros_(astronomy), http://www.hermit.org/eclipse/Graphics/diagrams/Draconic.png, http://www.metahistory.org/images/moonorbit.gif, http://www.astro.virginia.edu/class/whittle/astr1230/im/moon_sidereal.gif